From Saddle Points Toward Global Minima: A Newton-Type Method on Wasserstein Space
We study the minimization of non-convex functionals over the Wasserstein space. While recent work has showed that perturbed Wasserstein gradient methods can avoid saddle points for benign landscapes, existing approaches remain essentially first-order and do not provide fast local convergence once the iterates enter a neighborhood of a global minimizer. We propose Wasserstein Saddle-Free Newton (WSFN), a second-order method that preconditions the Wasserstein gradient by a regularized square root of the squared Wasserstein Hessian. This construction preserves attraction toward directions of positive curvature while inducing repulsion along directions of negative curvature, thereby overcoming the tendency of standard Wasserstein Newton dynamics to be attracted to saddles. We also establish second-order sufficient optimality conditions on Wasserstein space for strict local minimality. Under regularity and benign landscape assumptions, we prove that WSFN escapes saddle regions and reaches an $α$-neighborhood of a global minimizer in polynomial time, with improved dependence on saddle parameters compared with prior perturbed first-order methods. Once inside this neighborhood, we show that WSFN converges linearly in $L^2$-Wasserstein distance to a non-degenerate global minimizer. Finally, we present a particle-based implementation of the method.